Mathematics of apportionment


In mathematics and fair division, apportionment problems involve dividing a whole number of identical goods fairly across several parties with real-valued entitlements. The original, and best-known, example of an apportionment problem involves distributing seats in a legislature between different federal states or political parties. However, apportionment methods can be applied to other situations as well, including bankruptcy problems, inheritance law, manpower planning, and rounding percentages.
Mathematically, an apportionment method is just a method of rounding real numbers to natural numbers. Despite the simplicity of this problem, every method of rounding suffers one or more paradoxes, as proven by the Balinski–Young theorem. The mathematical theory of apportionment identifies what properties can be expected from an apportionment method.
The mathematical theory of apportionment was studied as early as 1907 by the mathematician Agner Krarup Erlang. It was later developed to a great detail by the mathematician Michel Balinski and the economist Peyton Young.

Definitions

Input

The inputs to an apportionment method are:
  • A positive integer representing the total number of items to allocate. It is also called the house size, since in many cases, the items to allocate are seats in a house of representatives.
  • A positive integer representing the number of agents to which items should be allocated. For example, these can be federal states or political parties.
  • A vector of numbers representing entitlements - represents the entitlement of agent, that is, the amount of items to which is entitled. These entitlements are often normalized such that. Alternatively, they can be normalized such that their sum is ; in this case the entitlements are called quotas and termed denoted by, where and. Alternatively, one is given a vector of populations ; here, the entitlement of agent is .

    Output

The output is a vector of integers with, called an apportionment of, where is the number of items allocated to agent i.
For each party, the real number is called the entitlement or seat quota for, and denotes the exact number of items that should be given to. In general, a "fair" apportionment is one in which each allocation is as close as possible to the quota.
An apportionment method may return a set of apportionment vectors. This is required, since in some cases there is no fair way to distinguish between two possible solutions. For example, if and, then and are both equally reasonable solutions, and there is no mathematical way to choose one over the other. While such ties are extremely rare in practice, the theory must account for them.
An apportionment method is denoted by a multivalued function ; a particular -solution is a single-valued function which selects a single apportionment from.
A partial apportionment method is an apportionment method for specific fixed values of and ; it is a multivalued function that accepts only -vectors.

Variants

Sometimes, the input also contains a vector of integers representing minimum requirements - represents the smallest number of items that agent should receive, regardless of its entitlement. So there is an additional requirement on the output: for all.
When the agents are political parties, these numbers are usually 0, so this vector is omitted. But when the agents are states or districts, these numbers are often positive in order to ensure that all are represented. They can be the same for all agents, or different.
Sometimes there is also a vector of maximum requirements, but this is less common.

Basic requirements

There are basic properties that should be satisfied by any reasonable apportionment method. They were given different names by different authors: the names on the left are from Pukelsheim; The names in parentheses on the right are from Balinsky and Young.
  • Anonymity means that the apportionment does not depend on the agents' names or indices. Formally, if is any permutation of, then the apportionments in are exactly the corresponding permutations of the apportionments in.
  • *This requirement makes sense when there are no minimal requirements, or when the requirements are the same; if they are not the same, then anonymity should hold subject to the requirements being satisfied.
  • Balancedness means that if two agents have equal entitlements, then their allocation should differ by at most 1: implies.
  • Concordance means that an agent with a strictly higher entitlement receives at least as many items: implies.
  • Decency means that scaling the entitlement vector does not change the outcome. Formally, for every constant c.
  • Exactness means that if there exists a perfect solution, then it must be selected. Formally, for normalized, if the quota of each agent is an integer number, then must contain a unique vector. In other words, if an h-apportionment is exactly proportional to, then it should be the unique element of.
  • *Strong exactness means that exactness also holds "in the limit". That is, if a sequence of entitlement vectors converges to an integer quota vector, then the only allocation vector in all elements of the sequence is. To see the difference from weak exactness, consider the following rule. Give each agent its quota rounded down, ; give the remaining seats iteratively to the largest parties. This rule is weakly exact, but not strongly exact. For example, suppose h=6 and consider the sequence of quota vectors. The above rule yields the allocation for all k, even though the limit when k→∞ is the integer vector.
  • *Strong proportionality means that, in addition, if, and, and there is some h-apportionment that is exactly proportional to, then it should be the unique element of. For example, if one solution in is, then the only solution in must be.
  • Completeness means that, if some apportionment is returned for a converging sequence of entitlement vectors, then is also returned for their limit vector. In other words, the set - the set of entitlement vectors for which is a possible apportionment - is topologically closed. An incomplete method can be "completed" by adding the apportionment to any limit entitlement if and only if it belongs to every entitlement in the sequence. The completion of a symmetric and proportional apportionment method is complete, symmetric and proportional.
  • *Completeness is violated by methods that apply an external tie-breaking rule, as done by many countries in practice. The tie-breaking rule applies only in the limit case, so it might break the completeness.
  • *Completeness and weak-exactness together imply strong-exactness. If a complete and weakly-exact method is modified by adding an appropriate tie-breaking rule, then the resulting rule is no longer complete, but it is still strongly-exact.''''''

    Other considerations

The proportionality of apportionment can be measured by seats-to-votes ratio and Gallagher index. The proportionality of apportionment together with electoral thresholds impact political fragmentation and barrier to entry to the political competition.

Common apportionment methods

There are many apportionment methods, and they can be classified into several approaches.
  1. Largest remainder methods start by computing the vector of quotas rounded down, that is,. If the sum of these rounded values is exactly, then this vector is returned as the unique apportionment. Typically, the sum is smaller than. In this case, the remaining items are allocated among the agents according to their remainders : the agent with the largest remainder receives one seat, then the agent with the second-largest remainder receives one seat, and so on, until all items are allocated. There are several variants of the LR method, depending on which quota is used:
  2. * The simple quota, also called the Hare quota, is. Using LR with the Hare quota leads to Hamilton's method.
  3. * The Hagenbach-Bischoff quota, also called the exact Droop quota, is. The quotas in this method are larger, so there are fewer remaining items. In theory, it is possible that the sum of rounded-down quotas would be which is larger than, but this rarely happens in practice.
  4. Divisor methods, instead of using a fixed multiplier in the quota, choose the multiplier such that the sum of rounded quotas is exactly equal to, so there are no remaining items to allocate. Formally, Divisor methods differ by the method they use for rounding. A divisor method is parametrized by a divisor function which specifies, for each integer, a real number in the interval. It means that all numbers in should be rounded down to, and all numbers in should be rounded up to. The rounding function is denoted by, and returns an integer such that. The number itself can be rounded both up and down, so the rounding function is multi-valued. For example, Adams' method uses, which corresponds to rounding up; D'Hondt/Jefferson method uses, which corresponds to rounding down; and Webster/Sainte-Laguë method uses , which corresponds to rounding to the nearest integer. A divisor method can also be computed iteratively: initially, is set to 0 for all parties. Then, at each iteration, the next seat is allocated to a party which maximizes the ratio .
  5. Rank-index methods are parametrized by a function which is decreasing in. The apportionment is computed iteratively. Initially, set to 0 for all parties. Then, at each iteration, allocate the next seat to an agent which maximizes . Divisor methods are a special case of rank-index methods: a divisor method with divisor function is equivalent to a rank-index method with rank-index .
  6. Optimization-based methods aim to attain, for each instance, an allocation that is "as fair as possible" for this instance. An allocation is "fair" if for all agents i; in this case, we say that the "unfairness" of the allocation is 0. If this equality is violated, one can define a measure of "total unfairness", and try to minimize it. One can minimize the sum of unfairness levels, or the maximum unfairness level. Each optimization criterion leads to a different optimal apportionment rule.